Cycle LDPC code

Cycle LDPC code[1]

An LDPC code whose parity-check matrix forms the incidence matrix of a graph, i.e., has weight-two columns.

The minimum distance of a cycle LDPC code is \(d\geq g/2\), where \(g\) is the girth of the code's Tanner graph [2; Remark 21.2.13].

Cycle codes are not asymptotically good [3].

Linear-time encoder [4].

Cycle LDPC codes have been proposed to be used for MIMO channels [5].

Quasi-cyclic LDPC (QC-LDPC) code — Cycle LDPC codes form a class of regular QC LDPC codes [6].
Regular LDPC code — Cycle LDPC codes form a class of regular QC LDPC codes [6].
Cycle code

Margulis LDPC code — Margulis LDPC codes are examples of cycle codes for particular large-girth graphs [7].

[1]: S. Hakimi and J. Bredeson, “Graph theoretic error-correcting codes”, IEEE Transactions on Information Theory 14, 584 (1968) DOI
[2]: C. A. Kelley, "Codes over Graphs." Concise Encyclopedia of Coding Theory (Chapman and Hall/CRC, 2021) DOI
[3]: L. Decreusefond and G. Zemor, “On the error-correcting capabilities of cycle codes of graphs”, Proceedings of 1994 IEEE International Symposium on Information Theory DOI
[4]: Jie Huang and Jinkang Zhu, “Linear time encoding of cycle GF(2/sup P)/ codes through graph analysis”, IEEE Communications Letters 10, 369 (2006) DOI
[5]: Ronghui Peng and Rong-Rong Chen, “Application of Nonbinary LDPC Cycle Codes to MIMO Channels”, IEEE Transactions on Wireless Communications 7, 2020 (2008) DOI
[6]: R. M. Tanner, D. Sridhara, A. Sridharan, T. E. Fuja, and D. J. Costello, “LDPC Block and Convolutional Codes Based on Circulant Matrices”, IEEE Transactions on Information Theory 50, 2966 (2004) DOI
[7]: G. Zémor, “On Cayley Graphs, Surface Codes, and the Limits of Homological Coding for Quantum Error Correction”, Lecture Notes in Computer Science 259 (2009) DOI

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“Cycle LDPC code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/cycle_ldpc